An Extension of Shelah's Trichotomy Theorem

In Sh506, Shelah develops the theory of $\mathrm{pcf}_I(A)$ without the assumption that $|A|<\min (A)$, going so far as to get generators for every $\lambda\in\mathrm{pcf}_I(A)$ under some assumptions on $I$. Our main theorem is that we can also generalize Shelah's trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for $\mathrm{pcf}_I(A)$ which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.